Bubble structures in acoustic cavitation

Robert Mettin

Drittes Physikalisches Institut, Universitat Gottingen,Friedrich-Hund-Platz 1, 37077 Gottingen, Germany

E-mail: R.Mettin@dpi.physik.uni-goettingen.de

RUNNING TITLE: Bubble structures in acoustic cavitation

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Abstract

This article is reporting on bubble structures that represent different manifestationsof acoustic cavitation. General aspects relevant for structure formation in acousticcavitation are discussed, and a classification scheme into prototypes is proposed.Characteristics and distributions of bubbles as well as thesound field environmentsare reviewed for the different cavitation patterns. The study is mainly based on op-tical and high-speed imaging investigations and is confinedto acoustic frequenciesof the lower ultrasonic range.

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1 Introduction

The rupture of a liquid leading to cavities of gas phase is encountered in differentsituations. Usually the circumstances are classified into the ideal cases of isobaricrupture due to heat deposition (boiling), and isothermal rupture caused by tension.The latter phenomenon, along with the subsequent dynamics of the created bubbles,is generally called cavitation [1]. It is furthermore subdivided according to thephysical origin of the tension: Rupture in a low or negative pressure zone of a bulkliquid flow is termed hydrodynamic cavitation [2, 3, 4, 5] , while the rupture instrong sound fields is named acoustic cavitation [6, 7, 8, 9, 10, 3, 11]. This article isdevoted to the visual appearance of cavitation in the lattercase - the generation byan acoustic field.The findings and results are restricted to a certain range of applied sound frequen-cies: the band between about 20 kHz and 50 kHz. The reason for this limitationis based upon both the importance of this range for technicalapplications, and thegood accessibility by optical sensors. Acoustic cavitation at higher ultrasonic fre-quencies creates increasingly small bubbles, and direct imaging becomes more andmore difficult. Then, indirect indicators have to be employed like erosion or sono-luminescence [2, 12]. Lower frequencies (in the audible domain and below) are notcommonly used, and the cavitation phenomena approach in a way hydrodynamiccavitation (oscillation periods are in the range of typicaltransit times of flows).Nevertheless, investigations of bubble dynamics in low frequency sound are as wellworthwhile and can reveal highly interesting results [13, 14, 15].The fields of applications of acoustic cavitation are permanently growing, and justas much growing is the demand for fundamental knowledge of the accompanyingphenomena. In most applications, like in ultrasonic cleaning [16, 17] and sono-chemistry [18], the collapse (violent implosion) of bubbles is the crucial eventwhich causes the observed effects. Consequently, the collapse phase is well studied,and much effort is undertaken to clarify the physical and chemical conditions of asingle collapsing bubble [9, 12]. For most kinds of experimental or technical setups,however, the link between applied sound field and bubble effects contains as wellthe distribution of bubbles in space and time. This distribution is usually found toappear in a given manner, and up to now it is not clear how to manipulate or controlit. A step towards this aim - a controlled acoustic cavitation - is a more completedescription and characterization of the existing bubble structures and their acousticenvironment. However, hitherto no satisfactory catalog ofacoustic cavitation phe-nomena exists, and this article tries to provide a starting or “nucleation” point forfurther works in this direction.It should be mentioned that a categorization of phenomena onthe side of hydrody-namic cavitation is already rather evolved. Owing to its importance in fluid dynam-ics engineering, the treatment of cavitation in flow problems has a long history (see,

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e.g., [9]). Starting from works related to ship propellers,researchers and engineersdeveloped an own vocabulary of the observed patterns. For instance, they refer totraveling cavitation, fixed or attached cavitation, supercavitation, vortex cavitation,and other manifestations of liquid rupture in flow problems [2, 9, 3, 5]. The mainaspect of distinction between the various types of hydrodynamic cavitation is thelocation of appearance and the visual impression.Quite in contrast, the existing categories of acoustic cavitation are led by qualita-tive features of the bubbles. Typical attributes are “transient” vs. “stable” or “hard”vs. “soft” cavitation. They describe the type of bubble collapse (transient = hard =violent collapse, stable = soft = collapse without high energy concentration [10]),but they do not qualify visual or structural bubble field properties or the environmentwhere the acoustic cavitation is observed. While the established notion is useful fora local description of cavitation, it clearly lacks the global aspect of bubble distri-bution in space and time. Vice versa, the categories of hydrodynamic cavitation donot tell much on the properties of single bubbles in the structures.The classical literature is using a relatively small vocabulary of the different visualmanifestations of acoustic cavitation. This seems to be partly owed to the sparse-ness of investigations, but also to the lack of sufficient imaging techniques. It hasbeen observed before that bubbles can appear in filamentary lines and conglomer-ates which have been termed “streamers” [8, 10], and also in cluster-like concen-trations [19, 20]. Additionally, some properties like subharmonic oscillations ofbubbles in structures have been visualized directly [21]. Apart of long-term ex-posures, however, the imaging by high-speed cameras or holographic systems hasbeen highly elaborate and expensive, and systematic investigations could hardly beperformed. This has changed considerably, and nowadays medium- and high-speedcamera systems are available at moderate cost, and fully digital image processingfacilitates subsequent analysis.

2 Prerequisites

This chapter contains general remarks on technical issues and on some fundamen-tals of acoustic cavitation which are relevant and useful for the discussion of struc-ture formation.

2.1 Experimental equipment

All of the following pictures have been recorded by several investigators at the ThirdPhysical Institute (Drittes Physikalisches Institut, DPI) at Gottingen University. Theimaging systems that have been used are a low framerate progressive scan videocamera (Pulnix TM 6701 AN), high framerate CCD cameras (HiSIS 2002, Photron

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ultima APX-RS), and an ultra high-speed camera with image intensifier (Imacon468). The cameras have been connected to suitable magnifying optics like zoomlenses or long distance microscopes.The illumination of cavitation processes is either arranged in a way that the bubblesscatter the light for recording (bright bubbles), or that they move in front of a ho-mogeneous bright background (dark bubbles). While the former way is somehoweasier to set up, it is hard to judge the absolute bubble sizesfrom the amount ofscattered light. In contrast, the latter method is suitableto obtain size statistics, butits resolution of very small bubbles can be worse: while the light scattered by a bub-ble could be enough to highlight a camera pixel, the shadowing effects of the samebubble in front of an illuminated background might not be sufficient for detection ifthe bubble image is smaller than a pixel. Likewise, the contrast of bubble signals inscattered light is typically better.The acoustic set-ups that have been used are mainly simple resonator systems,i.e. rectangular cuvettes with transparent walls, ultrasonic transducers at the bot-tom, and an open top. Then, the generated sound field is typically close to a stand-ing wave and consists of nodal and antinodal regions. Details of the field dependon the liquid filling height and other liquid properties, andon possible influenceof the emerging cavitation (shifting resonance frequencies by impedance changes).Additionally, the free surface can start to move and generate capillary and largersurface waves which can build up and cause swashing. This, inturn, can modulatethe whole sound field with low frequencies. The emitting transducer surfaces arerelatively large in our resonator baths, and the (average) intensity at the emitter islow (up to some W/cm2).Other types of set-ups contain sonotrode emitters (Mason horn transducers). Suchemitters typically have a small surface area, resulting in high intensity at the sono-trode tip in the range of 100 W/cm2. Structures will be shown near a small horn tipand others near a more extended one, which might be considered as a transition tolarger emitter surfaces like in the resonator baths. The sound field near a sonotrodecan have the characteristics of a running wave: due to strongscattering and dis-sipation in the strong cavitating zone directly in front of it, a shielding effect cantake place and the reflected waves from the container walls have less pronouncedinfluence than in the resonator case.All the structures reported in the following have been observed in water. Either thewater has been taken from the tap, or 20µm micropore filtered water has been used.In some cases the water has been degassed before the experiment. If non-degassedwater is used, its gas content decreases automatically under strong cavitating condi-tions, and during this initial process sheets of bubbly layers (“gaseous cavitation”)can appear. Later, “hard” (or “true”) cavitation remains orsets in. We always referto conditions after having run the experiments for a considerable time. Therefore,the actual gas content of the used water is typically undersaturated to a certain de-

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gree which, though, is not quantified.Acoustic cavitation is a complex phenomenon which involvesscaling in time andspace over several orders of magnitude. However, certain aspects occur on relativelydistinct scales. Typical durations of processes are 1 ns forlight emission and chem-istry at the end of a bubble collapse, 20 . . . 50µs for a bubble volume (and soundfield) oscillation cycle, 0.1 . . . 10 ms for the life and translation time of a bubblewithin a structure, 0.1 . . . 10 s for the change of bubble structures and sound fieldsby retroaction of cavitation or by influence of swashing. Typical dimensions are lessthan 1µm for the collapsed bubble, between 1 and 10µm for the equilibrium radiusR0 (the bubble size without sound field), 10 . . . 100µm for the expanded bubble,0.1 . . . 10 mm for bubble travel distances and up to some cm for structure sizes (therange of the sound wavelengths; all measures refer to the indicated frequency rangebetween about 20 and 50 kHz.).The separation of scales allows to investigate and describedifferent aspects individ-ually, which is an advantage for theoretical considerations. On the other hand, thebig span renders an all-embracing theoretical treatment difficult, if not impossible.Experimentally, it necessitates different recording equipment for capturing featureson different scales. The bubble patterns reported in this article typically live muchlonger than a single bubble oscillation period (which at allallows the notion of a“structure”). Their lifetime can be limited by slower rearrangements of the con-figuration of bubbles or sound fields, or can be also unlimited. At the same time,features like period-doubling can only be registered by recordings synchronous tothe rather short sound field period. The spatial dimensions of the structures in therange of mm to cm are as well very distinct from the size of individual bubbles.Therefore, the optical recordings presented in the following can refer to differenttime scales and spatial scales.

2.2 Sound fields

The characterization of the acoustic environment in the presence of cavitation isgenerally an involved issue. It is well known that the acoustic emission spectrumof cavitating liquids shows features like broad band noise and harmonics and sub-harmonics relative to the excitation frequency [22, 23, 9].While these spectralfeatures are essential for detection and characterizationof cavitation, they are cer-tainly induced effects and of only secondary importance forthe actual generationand structure formation of cavitation. Therefore, it is often valid to assume for the-oretical considerations that only the principal componentof the driving frequencyf =ω/(2π) is present in the sound pressure field which is varying sinusoidally intime t:

p(x, t) = pa(x) cos (ωt − φ(x)) . (1)

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The general notion of amplitudepa and phaseφ depending on the spatial coordinatex ∈ IR3 allows for standing or traveling wave components in the field. For example,

pa(x) = pa cos(k · x) , φ = const (2)

results in a plane standing wave, while

φ(x) = k · x , pa = const (3)

yields a plane traveling wave, both in the direction of a wavevectork and with awavelengthλ = 2π/k with k = |k|.As already mentioned, the assumption (1) is not perfectly valid in the case cavita-tion occurs in the liquid. Several effects lead to a retroaction of bubbles onto thesound field:1. The presence of bubbles changes the sound speed in the liquid (which is actu-ally a two-phase liquid-gas medium then). Accordingly, acoustic impedance andacoustic wavelength alter, and the radiation from the transducer as well as reso-nance frequencies of the cuvette can shift. Indeed, alreadya small void fraction ofgas can reduce the sound speed significantly (see, e.g., [24]), and recently the in-fluence of a single levitated bubble on the resonance frequency of a glass cube hasbeen measured directly [25]. Such changes can introduce temporal fluctuations ormodulations of amplitude and phase.2. The bubbles are created in the fluid and perform volume oscillations. This leadsto the loss and redistribution of energy of the primary wave:damping and scatter-ing. Therefore the amplitude of the pressure wave will be reduced when it traversesbubbly zones, and the phase can be changed. Amplitude loss will result in a highershare of traveling wave in the field (towards the regions of losses), and scattering toa certain randomization of the phase, which in turn leads to “speckles” in the soundfield that can vary in time if the bubble distribution changes.3. For stronger sound fields (and these are typically impliedfor cavitation), thebubble oscillations are nonlinear. This transfers energy to higher harmonics of thebasic frequency. Additionally, as mentioned, subharmonics and noise are generatedby the bubbles. The details of how exactly this happens are still partly unclear, butthe origin is necessarily nonlinear. Subharmonic (higher-periodic) oscillations ofbubbles and of whole bubble structures have been observed inexperiments [21, 26]and simulations [27, 28], and they are certainly linked to subharmonic lines. Alsochaotic (aperiodic) oscillations are shown in models, and there are hints from obser-vations that chaotic dynamics indeed can play a role. Furthermore, lifetime effectsof the bubbles broaden the spectral lines and are partly responsable for noisy back-ground.The effects listed above lead to a distortion or perturbation of the ideal monofre-quent sound field proposed above. Nevertheless it is mostly valid (and convenient)

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to neglect first of all any influence on structure formation. One subtle exception isthe secondary Bjerknes force being discussed later.The absolute values of pressure amplitudes shall be roughlydistinguished intolowwhich means in the range of 50 to 100 kPa and often lies below the cavitation thresh-old for the considered frequency range,mediumwhich denotes pressures between100 and 200 kPa, andhigh for pressure amplitudes beyond about 200 kPa. Notethat it is not possible to reach arbitrary high pressure values in the liquid. Cav-itation results in a certain saturation of the pressure amplitude because the liquidrupture impedes further tension being built up [19, 20]. Furthermore, shielding anddissipation effects of the bubbles limit the penetration depth of the sound.Part of the energy of the sound field is converted into acoustic streaming of the liquid[29]. The velocity of the liquid induced by acoustic forces has been monitored byobservation of submerged dust particles and by injection ofink. It turned out that(macro-)streaming reaches values in the range of 1 cm/s, which is often one order ofmagnitude smaller than the speed of cavitation bubbles. In many cases, acousticallyinduced liquid streaming does not affect the formation of bubble structures, whichis why we mainly disregard it in this context.

2.3 Bubble sources and cavitation thresholds

For the discussion of bubble distributions in acoustic cavitation it is important torealize that bubbles can be rapidly moving relative to the liquid, and that as well thesources of bubbles can migrate. The cause of bubble translation are acoustic forceswhich are theoretically quite well understood. They are called Bjerknes forces andare discussed briefly in the next chapter, and more elaborately in A. Doinikov’sSection of this Volume [30]. The behavior of bubble sources is theoretically lessclear, and much information has to be extracted and inferredfrom observations.Typically, bubbles appear at so-calledseeds(or nuclei). Only in very clean andpre-treated liquids and environments, rupture of the bulk liquid can be reached, andthen the necessary tension is mostly orders of magnitude higher than the experimen-tally observed cavitation threshold in “real” liquids [2, 1, 3]. Otherwise, cavitationinitiates at weak points in the liquid (the nuclei) which aregiven by pre-existing mi-crobubbles (of characteristic sizes below a micrometer), contaminations, containerwalls, object or transducer surfaces, or the pathways of high-energetic particle ra-diation [19, 31]. Some of these nuclei are fixed in space, but microbubbles andcontaminations can flow freely in the liquid. Furthermore, new populations of mi-crobubbles can be generated by other (larger) bubbles that split or disintegrate. Thespatial distribution of suchfree nucleiis difficult to observe in experiments becauseof their small size. Usually it is inferred indirectly by occurrence of larger, de-tectable bubbles.The difference between a (pre-existing) microbubble and a (sound-induced) cav-

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itation bubble is not extremely well defined. From a practical point of view, thecavitation bubble shows certain dynamics (in particular a strong collapse) whichleads to cavitation effects, while the nucleus bubbleper sedoes not. The distinctionis caused by the surface tensionσ which imposes the pressurepσ = 2σ/R upon theinterior of a spherical bubble of radiusR. Depending on the applied sound ampliudeand frequency, there exists a critical bubble radius below which the microbubblesremain inactive - i.e., they merely oscillate with small volume expansion, becausethe negative pressure phase of the sound field is not high enough to overcome thesurface tension [32, 3]. Beyond this threshold radius, bubbles can expand to a mul-tiple of their initial size and collapse violently afterwards [33]. For the case ofstatic tensionp < 0 in the liquid, and a liquid vapor pressurepv, the threshold wasgiven by Blake [32] to beRBl = −4σ/3(p − pv). Similar to a threshold radius,we can define a threshold pressure amplitudepBl for a given sizeR of the nucleus:pBl = pv − 4σ/3R. (The dynamical case of an oscillating pressure is discussed, forinstance, in [34].) This notion shows that a microbubble canturn into a cavitationbubble not only by increasing its equilibrium size, but alsoby translation when it en-ters a region of higher sound pressure. Accordingly, a cavitation bubble can becomea mere seed bubble if it shrinks or if the pressure falls belowthe Blake threshold.In addition, it can as well generate or transform itself intomany microbubbles bysplitting or disintegrating into small enough fragments, as mentioned above.Besides inactive microbubbles and stronger oscillating cavitation bubbles, there canalso exist larger, but only weakly oscillating bubbles in anirradiated liquid. Suchbubbles are typically near nodal zones of standing waves or close to container wallswhere the excitation pressure is low. They could be characterized as inactive (withrespect to strong collapse) macrobubbles. Nevertheless, they can serve as sourcesof migrating microbubbles by splitting them off [25]. Inactive macrobubbles canalso collect neighboring bubbles and grow large enough to feel a dominant buoancyforce which lets them rise to the surface. During degassing processes by ultrasound,as mentioned before, many such bubbles appear and play a certain role.The surface tension is not only causing the inactivity of very small bubbles. Evenmore, theoretically it forces small bubbles to dissolve because of the higher gas pres-sure inside [35, 9]. The experience shows, however, that in real liquids free nucleiexist even after long rest times. Thus one concludes that themicrobubles are stabi-lized against dissolution, for instance by liquid contaminations (surface active sub-stances gathering at the bubble’s surface) or by freely floating solids (e.g. bubbles inmicroscopic cracks of tiny particles). When a sound field is acting, the oscillationof the bubbles can as well counteract the dissolution process. This effect is calledrectified diffusion(RD) and means that the net flux of gas through the bubble wallafter one oscillation cycle is not zero. Increased surface area, nonlinear oscillationand the so-called “shell effect” contribute to a greater influx during the expandedbubble phase than there is outflux during the collapse phase [36, 9, 37]. Again, this

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process shows a threshold: For too small bubbles, the oscillation is too weak andsurface tension prevails: the bubble should dissolve. It has been speculated that therectified diffusion is the dominant process which turns inactive microbubbles intostrongly collapsing ones, and therefore the RD threshold ispartly considered as athreshold for cavitation at all. However, recent calculations show that for increasedpressure and for small bubbles the RD threshold more or less coincides with theBlake threshold [38], at least for the lower frequency ultrasonic range. Thus, verysmall bubbles are either dissolving, or active (strongly oscillating), and there is onlyvery few parameter space for (inactive) microbubbles growing by rectified diffusion.Alternatively, the process of translation, collison and merging may be dominant formicrobubble growth in many parameter regimes [39]. A deeperdiscussion of thisidea which would imply that merging works faster than dissolution results in certainestimates of microbubble densities and will be given elsewhere.Still another threshold relevant for acoustic cavitation phenomena is the transitionto aspherical bubble shape [40, 41, 9, 10, 3]. Depending on frequency and bub-ble equilibrium size, there exists always a sound pressure amplitude beyond whichthe spherical shape of an oscillating bubble is unstable with respect to small per-turbations. This threshold is generally lower for larger bubbles, but it shows reso-nances where eigenfrequencies of surface modes fit to rational multiples of the driv-ing frequency. Then the instability is easier triggered. Inessence, at fixed drivingfrequency, thissurface instabilitythreshold limits the allowed bubble equilibriumradius, depending on the local pressure amplitude. Bubblesbeing larger than thestable size are going to split or disintegrate.

2.4 Bjerknes forces

The acoustic forces acting on and between oscillating bubbles are called Bjerknesforces [42, 10]. Their origin is a nonzero average of the instantaneously acting forceF(t) = −V (t)∇p(x, t) which is felt by the bubble at any instant of time under themain assumption that the bubble at positionx is small compared to the wavelengthof the sound field (see [30]). Supposing both bubble volumeV and pressurep areoscillating periodically, we can time average over the periodT to arrive at

FB = −〈V (t)∇p(x, t)〉T . (4)

For simplicity we keep the indicated assumptions (more about the excluded caseslike larger and shape distorted bubbles can be found in [30] and the literaturetherein). If the pressure fieldp is caused by an external source (typically the domi-nant field), the force is calledprimary Bjerknes force, and if the field origins fromneighboring bubbles, it is termedsecondaryBjerknes force.

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For the monofrequent pressure distribution Eq. (1) we find

FB1 = −∇pa(x) 〈V (t) cos (ωt− φ(x)) 〉T+ pa(x)∇φ(x) 〈V (t) sin (ωt− φ(x)) 〉T . (5)

The first term on the right hand side can be identified as contributing for standingwaves, and the second for traveling waves. Forces are exerted in the direction of thepressure amplitude gradient, and the phase gradient, respectively. The sign of thepartial forces depend on the phase relation of the bubble volume oscillation to thedriving pressure (cosine) and its derivative (sine). For small oscillations (linearizedmotion of the bubble wall) the common notion is found where the linear resonanceradiusRres [10] plays a significant role: In standing waves, bubbles smaller thanRres travel to pressure antinodes, while bubbles larger thanRres go to nodes. In atraveling wave, the force is always in direction of the wave propagation, vanishes forvery small and very large bubbles, and is largest forR = Rres. If nonlinear bubbleoscillations are taken into account, it is found that also small bubbles are repelledby pressure antinodes for sufficiently high pressure [43, 44], and that bubbles intraveling waves can also run backwards [45].A detailed discussion of the secondary Bjerknes force can befound in the literature[10, 30]. Here we want to note that, to a certain approximation which is sufficientfor many aspects of bubble structures, the force of bubble “1” at positionx1 exertedon bubble “2” at positionx2 can be written as

F1,2

B2= −

ρ

4π

(x2 − x1)

|x2 − x1|3

⟨

V1(t)V2(t)⟩

T. (6)

Hereρ is the liquid density, and the dot denotes time derivatives of the bubble vol-umesV1 andV2. Important features are that the force deminishes with the squaredbubble distance and that it acts typically attractive for bubbles of similar size whichare close to each other. However, many complications and details can be found fornonlinearly oscillating [46] and shape distorted bubbles,very close encounters etc.(see again [30] for a review).It should be remarked that, in a way, the inclusion of the secondary Bjerknes forceis inconsistent with our previous assumption of the (primary) sound field beingundisturbed by the presence of bubbles. This can partly be justified by the fact thatthe contributions scattered by the bubbles decay over a short range and act onlylocally.Generally speaking, the Bjerknes forces lead to a spatial concentration of bubbles.Primary forces drive the cavitation bubbles towards pressure antinodes in standingwaves and larger (degassing) bubbles to nodes. Neigboring bubbles tend to clusterdue to secondary forces. Furthermore, acoustically hard object surfaces can attractbubbles due to secondary Bjerknes forces from reflections (virtual mirror bubbles).

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3 From the zoo of bubble structures

This chapter provides a catalogue of observations and, in some cases, simulationsof bubble structures. The classification has been guided mainly by visual aspects,and it has to be read as a suggestion to group and term the various manifestations ofacoustic cavitation.

Figure 1: One-dimensional streamer filament: Bubbles originate nearthe top at a needletip (arrow), and they run downwards in a straight streak to stop in a small cluster (at thebroadened end). Recording in scattered light, exposure time 16 ms (from [25]). All whitespots aside the central vertical strip are dust particles inthe liquid and no bubbles.

3.1 Streamer and filament

An important building block of acoustic cavitation structures is astreamerwhich isshown in its “pure” form in Fig. 1. It consists of a linear streak of bubbles that travelrapidly from one end to the other, forming a directed bubble stream. The figureshows a streamer in specially prepared environment where the bubbles are artifi-cially seeded at the tip of a needle [25]. Often the sources ofthe streaming bubblesare more obscure because the bubbles are first too small to be visible. Then the ap-parent starting point of the streamer is located somewhere in the liquid. Typically,however, a closer microscopic inspection reveals that manymicrobubbles already

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(a) (b)

Figure 2: Two examples of filamentary streamer configurations (Acoustic LichtenbergFigures). Recordings at 25 kHz in scattered light, but with inverted grey scales (thereforebubbles appear dark). (a): longterm exposure (20 ms) in a cylindrical resonator cell, pictureheight≈ 2 cm; (b): short exposure (5µs) in a cubical cell, dimension≈ 1 cm, from [47]).

form a stream to the apparent source point. The actual sources can be containerboundaries or really free floating microbubbles gathering at the virtual streamer ori-gin. The end point in Fig. 1 is a small cluster of bubbles. In more natural casesstreamers merge with other streamers, and many of them form an agglomerate offilaments. This can be seen in Fig. 2 where typical filamentary bubble structuresare given. Bubbles move from outer regions towards the center and finally reacha central point of high bubble concentration. The whole structure resembles rootsof a plant, neurons, or electrical discharge patterns. For the latter reason, filamen-tary structures like in Fig. 2 have also been termed “Acoustic Lichtenberg Figures”(ALFs) [48]. For only inward traveling bubbles in a filament,the question of con-servation of gas mass arises. It seems that bubbles at the central cluster finallydisintegrate into microbubbles and dissolve or are taken away by liquid flow. Thelatter is possible for small enough bubbles as they don’t feel a strong Bjerknes force.Indication for this scenario is a fuzzy mist or halo around the center in some record-ings like Fig. 2(a).The acoustic environment typical for streamers are standing waves. Bubble sourcepoints are close to lower pressure amplitude regions (nodesor walls), and the di-rection of bubble translation is towards higher amplitudes, in particular towards thepressure antinodes. The absolute pressure values at the antinode typically do not ex-ceed 180 kPa, otherwise it repells the bubbles [43] and they cluster farther outside.Therefore, streamers are mainly found for weakly cavitating systems just beyondthe cavitation threshold or in zones of low to medium pressure amplitudes. If a

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Figure 3:Comparison of measured and reconstructed three-dimensional bubble path data(lines) and the results of a particle model calculation (crosses). The model creates for eachdetected bubble a “particle” that subsequently follows theequations of motion accordingto Bjerknes forces. The generated tracks are relatively close to the real ones (from [52],simulation parameters: 30 bubbles ofR0 = 5µm, driving 150 kPa at 20 kHz).

central filament is driven too strong, it rearranges to an off-center filament [48, 49].The main mechanism behind streamers and filaments can be explained by the Bjerk-nes forces. Small bubbles which are generated near nodal zones feel attraction to-wards the antinodes by the primary Bjerknes force. Clustering of microbubbles nearthe (virtual) origins of streamers is mediated by the secondary Bjerknes force. Thesame is true for longitudinal interaction along the path, and for merging of filamentbranches. This process is relatively well understood up to quantitative consistency.Simulations by a particle model [50, 51] which take into account Bjerknes forcesand a suitable viscous drag force could reproduce three-dimensionally reconstructedbubble tracks very well [52, 53, 54]. An example from Ref. [52] is shown in Fig. 3.Streamer filaments have been investgated quite well. From holographic and stereo-scopic recordings [21, 55, 52] it has been possible to reconstruct three-dimensionalpictures and bubble motions. Furthermore, bubble sizes andvelocties have beenevaluated [56]. The main results are that the equilibrium radii of the bubbles staybelow about 10µm, and that their speeds reach up to about 1 m/s (with a distribu-tion peak around 0.3 m/s [57, 55]). Larger and faster bubblesare seldomly found.Bubble densities within the streamers are smaller than one expects from a long-term exposure like Fig. 2(a). Typically bubbles don’t have lateral neighbors, andthe distance between subsequently traveling bubbles amounts about 0.5 to 2 mm.

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1 cm

Figure 4:Streamer of curved shape (bow). Recording in scattered light at 25 kHz, exposure0.3 ms, inverted grey scales.

It is important to notice, however, that the bubble stream can be rather discontinu-ous and that bubbles can merge and split within the track. Consequently, there is alongitudinal interaction between bubbles, and bubbles cantemporarily stop or runbackwards to merge with another one.A striking feature of streamers is their filamentary arrangement to ALFs. The spe-cific filament branches possess a remarkable high stability and long lifetime, whichis one reason for inhomogeneous spatial bubble densities even for longer time aver-ages. In simulations, this could be reproduced by the assumption of an inhomoge-neous distribution of bubble sources fixed in space like at walls or objects [48, 49],together with secondary Bjkerknes force effects. However,it stays unclear why thefilaments are rather stable in the free liquid as well where the nuclei are supposedto float arbitrarily in space. One can speculate that the microbubbles gather in aself-amplifying way based on the instability of a homogeneous distribution [48].Furthermore, self-stabilization of the bubble branches could be possible by the bub-bles serving as sources themselves (by splitting off microbubbles along their path,which in turn “feeds” the branch). Such details are still to be clarified.

3.2 Bow and ring

Streaming flows of bubbles can also appear in a curved fashion. Figure 4 shows anexample of a bow which is ending in a one-sided filament near the left border of thepicture. The structure is actually three-dimensional, andwe can see more than onearc terminating in the same cluster (due to a lack of depth of focus, the respective

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(a) (b)

Figure 5:Rings of streaming bubbles at 25 kHz (scattered light, exposure 0.8 ms). Pictureheights approx. 4 cm.

bubbles appear blurred). In some cases, the bubble sources could definitely iden-tified as attached to the container walls, in other cases (like in Fig. 4) the sourcesremain unclear as for “free” streamer sources. It seems evenpossible to have closedrings when the end cluster (or filament) of one arc appears to “feed” another oneand vice versa. This case is presented in Fig. 5. However, it remains to be shownthat indeed bubbles can travel all around the circle and thatit is not composed oftwo half-circles.Typical acoustical environment is a standing wave field at small to moderate pres-sure amplitudes. Hydrophone measurements show that the curved bubble tracksencircle pressure nodal zones and lie close to antinodal regions. It seems to beessential that the standing wave is more extended, i.e. it contains more than oneantinode. The absolute pressure range is similar to that of other filaments. Thisindicate also the pressure measurements of Hatanaka et al.,who as well observeda ring-like structures in their study [58] and report pressure node amplitudes below100 kPa.For increased pressure the ring structure can tranform to the double layer structurewhich is described in the next section. This can already be anticipated in Fig. 5(b).There the excitation is higher than in Fig. 5(a), and two parts of each ring showstronger bubble population.The following mechanism is likely to explain the structure:Similar to straightstreamers, the bubbles run, driven by the primary Bjerknes force, from low-pressureto high-pressure regions. The standing wave field is formingsomething like moun-tains where the bubbles try to reach the (pressure amplitude) peak. In contrast to asimple central antinode and a direct way upward like before,here the wave patternis more involved and provides a curved path. Furthermore, only few bubble sourcesseem to be contributing. From this picture, closed rings should not be possible,unless there is a mechanism for a bubble to “fall down” from a peak. Again, disin-tegration into microbubbles might be a way for the gas to escape from the antinode,following an outward (“downhill”) stream of fluid.

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(a) (b)

Figure 6: Double layer structure from the side (a) and from the top (b).Scattered light,25 kHz, long-term exposure, picture heights about 4 cm (from[59]).

The bubble sizes, velocities and densities are similar to straight streamers and otherfilaments. The main difference appears to be the detailed form of the standing wave,but apart of this the phenomenon is just another form of a streamer.The necessary properties of an extended standing wave field for curved and ring-shaped streamers remain to be shown.

3.3 Double layer (jellyfish)

This structure consists of two layers of bubbles which appear flat or slightly curvedfrom the side, but like filamentary stars from the top. Both perspectives are shownin Fig. 6 as seen by a long-term exposure. The layers seem to bealways existing inpairs, are often of similar extension, but can as well occur unbalanced in size. It ispossible that a layer stretches perpendicular to its planardirection to form a ratherconical entity. The double layer structure has been termedjellyfish[59, 60] becauseits fuzzy and slightly drifting appearance bears some similarity to these transparentand floating animals. It has been shown that the structure hasstrong cleaning anderosive potential at objects [61].Closer inspection by microscopic high-speed recordings reveals a very dynamicnature of the layers. Small bubbles stream into the layers from the region betweenthem, and larger bubbles or clusters form within the layers and move in the oppositedirection. At times, a large stable bubble is found between the layers.The structure exists in well defined standing waves at high pressure amplitudes. Thetwo layers arrange symmetrically with respect to a nodal pressure plane. Accord-ingly, their bubbles oscillate in antiphase: While the bubbles of one layer are neartheir maximum extension, the bubbles of the other layer are at their collapse, andvice versa. This type of alternating collapse can be seen in Fig. 7.A jellyfish structure can move a considerable amount in parallel to the nodal plane,

17

12.5 us 18.75 us

43.75 us37.5 us

6.25 us

31.25 us

0 us

25 us

Figure 7: Antiphase oscillation of a double layer structure (drivingfrequency 40 kHz,backlit exposures of 1µs, frame width 1 cm; from [60]).

but usually never traverses it. Therefore, structures do not interact directly withperpendicular neighbor structures, but can and do interactwith other jellyfishes inthe same plane; sometimes they split into two, or two neighbors merge into onestructure. A transfer of bubbles across antinode regions takes place as well, butonly in form of fast moving larger clusters of bubbles.The lifetime of a jellyfish structure can be very long, but in many cases it is limitedby some interaction with neighbors, or by deterioration or alteration of the standingwave. For example, a free surface of the liquid can influence the acoustic field con-siderably by swashing or surface waves, in particular at higher frequencies wherethe wavelength is shorter. This in turn can change or destroy/rebuild the doublelayers in the rhythm of the swashing. At the same time, the noise level producedby the cavitation is changing. Indeed, it has been shown thatthe emergence of thejellyfish structure comes along with the subharmonic [25, 62].Most bubbles appear from the region of the nodal plane between the layers. Some-times the bubble stream is partially fed by a larger bubble which is trapped at thenode, but usually it is not clear from where exactly the bubbles originate, similarto free origins of streamers. Due to primary Bjerknes forces, the bubbles traveltowards the antinode. However, the agglomeration within the layers is not at theantinode itself, but somewhere before: The high pressure amplitude at the antinodeleads to repulsion of the bubbles, and the bubble layers settle near the location ofzero primary Bjerknes force. On their way towards and into a layer, the bubblesagain tend to form filaments because of the secondary Bjerknes force attraction.This process has been simulated qualitatively by a particlemodel [60] which showsthat basic features can be reproduced. As an example see Fig.8 which has beencalculated by P. Koch (DPI).From Ref. [60] we cite some statistical data for jellyfishes at 40 kHz driving: A layerconsists of around 1000 visible bubbles, bubble number densities reach 20 mm−3

within the filaments, maximum bubble radii are in the range of20 to 50µm, and

18

Figure 8: Particle model simulation of a double layer structure: sideview (left) and topview (right). Simulation parameters: up to 700 bubbles ofR0 = 3...5µm, driving 200 kPa at25 kHz.

Figure 9:Subsequent frames from a synchronous high-speed recordingof a jellyfish struc-ture (backlit, exposure 1µs, 26 kHz driving frequency). The interframe time amounts ex-actly one driving period. In the filaments, a period-2 breathing of the structure can be seen.

bubble velocites reach 2 m/s. The actual pressure amplitudeat the layer should bearound 200 kPa for a Bjerknes force balance, while the antinode pressure can beconsiderably higher. The respective equilibrium radii of the contributing bubblescan be recalculated to amount a few micrometer or less.An intriguing feature of the jellyfishes is the connection with subharmonic emis-sion. By synchronous recordings of images and acoustic spectra for a slowly mod-ulated pressure amplitude it has been shown in Refs. [25, 62]that apparently onlythe structure formation leads to emission at half the excitation frequency, while “un-structured” cavitation (appearing in small single streamers) did not. The limitationsof magnification and time resolution in this work prevented adirect observation ofsubharmonic bubble oscillations, but the phenomenon of a collective higher peri-odic oscillation of a whole structure has been observed before for a larger filament[21] and recently for large bubble clusters that we will discuss later [26]. Now

19

(a) (b)

Figure 10:Two configurations of a sub-surface bubble layer (starfish).Recording at 25 kHzin scattered light, longterm exposure, frame width about 2 cm.

we present a high speed recording sequence which demonstrates a period-doublingfor a jellyfish directly. The frames of Fig. 9 have been recorded exactly with atime interval of the driving period at 26 kHz. Clearly a period-two “breathing” ofthe structure is visible. What exactly leads to this structural period-doubling is notclear yet. Synchronization between bubbles [63, 64] and/orglobal interaction of thestructure with the sound field are possible options.

3.4 Surface bubble layer (starfish)

While bubble layers of a jellyfish always appear in pairs, it is possible to find asingle bubble layer just beneath a free surface of the liquid(i.e., under the liquid-airinterface). This configuration is calledstarfish[25], and an almost circular exampleis shown in Fig. 10(a).The structure appears at the same acoustic conditions as thejellyfishes, and indeedit corresponds mainly to one half of a jellyfish. The free surface builds a nodalplane in the standing wave, and the bubbles form the highest possible layer withouta partner layer, which would be outside the liquid. Nevertheless, there are at leasttwo specialities of this structure which render it distinctand which justify an ownname for it. First, its stream of bubbles (which is again fromthe node towards theantinode) can be fed from the free surface. Air can be entrained, apparently throughsurface waves or capillary waves. Therefore bubble sourcesare abundant, and thefilamentary layer frequently has a larger extent than the double layers within the

20

Figure 11:Examples of small cluster configurations at 25 kHz (courtesyS. Luther [47];frame sizes about 1 mm, backlit exposure about 2µs).

bulk liquid. Second, the surface can show a pronounced outward bulge right abovethe starfish. It is not clear what causes the bulge, but it appears similar to the acousticfountain effect [65]. Possibly there exists a self-amplifying interaction between thestructure and the convex surface bulge, which in turn deforms the standing wavelocally.Depending on the conditions of the resonator bath, starfishes can live long and staymore or less stationary, or show a dynamic emergence and disappearance with life-times in the range of seconds. Similar to jellyfishes they canmove parallel to thesurface and collide with neighbors, in which case sometimesan elongated layerfilament with a pronounced “backbone” forms. This is shown inFig. 10(b).The creation of the structure is similar to one jellyfish layer, but there are no definitedata about bubble sizes, densities and velocities up to now.Visually the structureoften appears larger and more pronounced than a jellyfish, and possibly bubbles arebigger and more dense due to air entrainment. An interestingapproach has been car-ried out by Krefting and Parlitz [25] to characterize the distinction between starfishconfigurations and streamer filaments (Sec. 3.1) by a fractaldimension. Aware ofthe limitations of the applicability of the method to the 2d-image data, they find aslightly larger dimension for the starfish, which reflects the more dense and morefiligrane filaments compared to the streamer conglomerations (compare Fig. 2).

3.5 Cluster

Compact groups or agglomerations of many bubbles will be termedcluster in thefollowing. Clusters can be isolated, i.e. without visible inflow of bubbles or fila-ments, or can be located at the ending of one or more streamers. They can existfree in the liquid or attached to surfaces, for instance at submerged objects. Atleast two types of clusters, characterized assmall and large, can be distinguishedphenomenologically. This is expounded in the following.

21

Figure 12:Variability of a small cluster: every exposure has been taken at fixed phase atsubsequent driving periods at 24 kHz (from [25]; frame sizesabout 1 mm, backlit exposure1µs).

3.5.1 Small cluster

A small cluster consists of only a few up to some dozens of visible bubbles whichare localized within a region of about 0.2 to 0.5 mm diameter [47, 25]. With thenaked eye it can appear like a larger single, jiggling or dancing bubble, but with suf-ficient magnification and short exposure time, individual bubbles within the smallcluster are clearly identified, see Fig. 11 for examples. Thebubbles show a strongoscillation behavior, and during their collapse phase theycan shrink below opticalresolution to disappear from the image. While near their maximum, the individualbubbles have distances comparable to their own size, but they can also touch eachother and partly merge. The structure in the cluster is extremely variable, as canbe seen from Fig. 12: Not only the positions of the bubbles canchange during oneoscillation cycle, but also their number. The cluster as an entity, in contrast, is quitestable and long living. It shows certain features of a singlebubble, like the reactionto Bjerknes forces: small clusters can be levitated in a standing wave field, theycan move and they feel attraction from other clusters in the liquid. Their seedingcan take place via a streamer. For instance, the single streamer filament from Fig. 1ends in a small cluster. When the bubble inflow from a streamerstops, the clustercan disappear, but it can also persist isolated from (visible) bubble sources. Theexample from Fig. 12 has been recorded from a levitated, longliving and isolatedcluster.Small clusters appear for elevated pressure amplitudes above about 190 kPa andbelow about 300 kPa [25]. Too strong excitation destroys thecluster or makes it

22

Figure 13:Motion of a spontaneously emergent large cluster towards a glass plate. Back-lit, non-equidistant frames from a high-speed series, standing wave at 40 kHz, timing andspatial scale indicated (picture from [25]).

drifting to the walls. All recorded examples stem from standing wave fields in the20 kHz range.Measurements show maximum radii from 15 to 65µm, and equilibrium radii ofroughly 1µm can be deduced from this. Furthermore it was found that theydo notemit subharmonic lines, and that with respect to Bjerknes forces they behave similarto single bubbles of about 10µm equilibrium radius [25].It is unknown why the structure is stable, while at the same time being extremelyvariable in its constituents. Naively thinking, the bubbles should attract each otherand coalesce, because they oscillate in phase which causes an attractive secondaryBjerknes force. However, it is not definite how the secondaryBjerknes force acts formore than two bubbles oscillating strongly on such small distances. Probably, somecomplications from a near field interaction come into play [66, 30] which modifythe simple attraction scheme. Additionally, a larger (merged) bubble is likely to beshape unstable and would probably split. It seems that the structure finds a “compro-mise” between merging and splitting. From recordings it is evident that there is notmuch translation of the bubbles during their phase of extended volume, but theremight be a rapid spatial motion during collapse, not visibleon the pictures. Re-cent particle simulations by Konovalova et al. including nonlinearity, coupling andtranslation of a few bubbles [67] show that for small clusters stable (non-merging)parameter regions can exist. On the other hand, there is a remarkable similarity ofwhat we call a small cluster to the early reports and picturesof surface unstablebubbles by Kornfeld and Suvorov from 1944 [40]. Their images, taken at 7.5 kHzsound frequency, show a distorted bubble that at times splits into smaller parts, butat other instants seems intact. Possibly the transition from a highly deformed bubblethat rarely splits off parts to a more or less permanent smallgroup of individuals isgradual and parameter dependent, and the small cluster is one extreme case.

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0.5 mm

1/4 T

5/4 T

2/4 T

6/4 T

3/4 T

7/4 T

4/4 T

8/4 T

Figure 14:Time resolved oscillation of the bubbles inside a large cluster which has beenstabilized in its position by a needle tip. Recording at 40 kHz, timing given in units ofT = 25µs (backlit exposure 2µs).

3.5.2 Large cluster

Large clusters are composed of up to hundreds of individual bubbles, much morebubbles than in the small clusters. They show maximum radii and mutual distancesin the range of their small cluster counterparts, but the agglomeration itself extendsup to the millimeter range. Large clusters can appear spontaneously in the free liq-uid where they usually move rapidly in space. Surfaces can attract large clustersand afterwards stabilize them in position and time. The shape of the structure is attimes surprisingly spherically (or hemispherically on surfaces), but it is evident thatit is, unlike the small cluster, far from a single merging/splitting bubble. Neverthe-less large clusters are subjected to Bjerknes-like forces as entities and show otherproperties of larger single bubbles. An example of a large cluster and its motioncan be found in [25] and is reproduced in Fig. 13. Here, the individual bubbles canbe optically resolved, and the spherical shape in the free liquid is nicely seen. Thecluster reaches the surface of a glass plate and evolves intoa rather hemisphericalshape. During this transition, it flattens first, as having aninertia, and then reshapesinto the hemisphere as if under the influence of surface tension.Large clusters have been observed in standing wave fields at elevated pressures un-der similar conditions where jellyfish structures appear. They can travel and trans-port bubbles perpendicular to the standing wave planes, other than jellyfish layers.Indeed, this perpendicular direction seems to be preferredby large cluster motion,at least according to traces of surface cleaning and erosion[61].The mechanism of the structure is not clear. Similar to smallclusters all bubblesshould attract each other and merge, because they oscillatemore or less in phase.This can be seen from very high-speed images like the one in Fig. 14. Two com-plete driving periods have been covered by the eight frames of the recording, andthe periodicity and the strong collapses of the constituting bubbles are resolved.

24

There is a quite high local variability in the structure. Thelifetime of individualbubbles in the large cluster seems to be very short as in the small cluster version,but sometimes individual bubbles can be observed for some subsequent cycles [26].Another characteristic feature of large clusters is shown in Ref. [28]: often theytend to period-doubled or higher periodic oscillation which is reflected in the visi-ble extension of the whole structure. Accordingly, subharmonic emissions can begenerated, similar as for jellyfish structures. This collective period-doubling effectseems to be a feature connected with the bubble number in the agglomeration, andit has been investigated on basis of a simple cluster model [28]. One result is thatindeed period-doubling can be facilitated by increasing the bubble population, if thebubbles are not too small.Larger clusters on surfaces frequently look like attached hemishperes, and theyshow a high erosive potential. There are models of shock waveamplification insuch a cluster [68] by a subsequent collapse of concentric shells, and this wouldcontribute to strong erosion. However, to the author’s knowledge this phenomenonhas not been observed or proven experimentally yet.Earlier high-speed observations of clusters are reported in Rozenberg’s book [7]and by Crum and Nordling [69]. Rozenberg’s images show sequences of a largercluster attached to a surface, and Crum and Nordling show stroboscopic images offree spherical large clusters with high translating speed (which they callcomets).Although their work was done in focused fields, there are obvious similarities oftheir pictures to our recordings of larger clusters, and it is likely that they observedsimilar structures.

3.6 Sonotrode cavitation

Transducers of the form of a vertical bar dipping into the liquid are called sonotrodesin the following. Typically they have a small emitting surface with accordingly highintensity (many W/cm2). They are frequently used for applications in the 20 kHzregion like disintegration, homgenization or sonochemistry. The transducer oscilla-tion is transformed to high amplitude by a mechanical amplifier in form of a horn[70]. The resulting radiating surface is usually much smaller than the acoustic wave-length, and a form of a propagating sound beam is created (piston source [71]).Further influence on the acoustic field is given by the shape and size of the liquidcontainer which causes standing wave contributions, but often these are negligible.The sonotrode cavitation is specific for small and intense emitters of progressivewaves. Bubbles appear predominantly in front of and close tothe sonotrode tip, andthe cavitating zone extends with increased transducer power. Detailed pictures ofbubbles at and near a sonotrode surface have been published already by Kornfeldand Suvorov [40] and by Olson and Hammitt [72]. The authors were investigat-ing the erosion mechanism of cavitation (see also [2]), and the pictures reveal an

25

Figure 15: Examples of bubble configurations below a sonotrode of 1 cm diameter at24 kHz (backlit, exposure time 2µs, picture height about 1 cm). The sonotrode tip is visibleat the top of the pictures, and three arbitrary, but increasing intensities are given from left toright.

inhomogeneous coverage which sometimes appears star-like.Examples of bubbles in the liquid from a side view are shown inFig. 15. Increasingthe power stretches the bubbly region, but also raises the bubble density and seemsto enlarge the bubbles themselves. The sonotrode surface ismore and more denselycovered with large bubbles.The bubble distribution in the liquid looks more or less homogeneous, and con-glomerations or filaments are not prominent. Sometimes, streaks of bubbles appear,but overall the structure is relatively featureless and smooth. Also, the bubbly zonecan be bending to the side, deviating from a vertical extension of the sonotrode.Bubbles seem to be mainly created close to or at the sonotrodetip, but some canalso appear from the liquid moving towards the center of the cavitation zone. Thelifetimes of recorded bubbles range from one or a few acoustic cycles (at locationsclose to the sonotrode tip) to many cycles more far. The distant and longer livingbubbles can be tracked to find their motion velocity and direction. Often bubblesmove fast away from the tip (about 1 m/s), but also bubbles moving towards the tiphave been observed [73]. The bubble translation can be theoretically explained byBjerknes forces in traveling waves [53]. In this case, the phase gradient term inEq. (5) is of great importance. There is always a strong liquid streaming inducedby sonotrode transducers, and the liquid flow appears to be responsible for at leastpart of the structure shape. When the flow is hindered artificially [74] or by largerextension of the sonotrode surface, the bubbles form a conical zone, see the nextSection.Statistics of bubble distributions below a sonotrode have been reported by Burdin etal. [75]. They find mean bubble radii below about 10µm and a homogeneous spatialdistribution below the tip. Additionally, void fractions have been determined.It has been reported that cavitation effects can differ significantly between small,high intensity sonotrodes and larger emitting surface setups [76]. This might beconnected with the bubble structures and in turn with the bubble densities, lifetimes

26

(a) (b)

Figure 16: Bubbles below a large sonotrode in scattered light: (a) long-term exposure(width 15 cm), (b) frame from a high-speed recording (exposure time 0.88 ms, width 6 cm).The sonotrode has a diameter of 120 mm which corresponds to 1.64λ for the driving fre-quency of 20500 Hz (pictures from [78]).

or sizes. Much is still unknown with respect to this aspect, but it is a main problem inupscaling acoustic cavitation processes. Some data has been published recently, butmore detailed investigations of sonotrode bubble distributions are still in demand.

3.7 Conical bubble structure

The conical bubble structure (CBS) appears at elevated intensities under a sonotrodeof large emitting surface with a diameter in the range of the wavelength [77, 78]. Itsshape is shown in Fig. 16(a) as it appears to the naked eye. Short-term exposuresreveal the composition of streaming bubbles in filaments that origin at the transducersurface and reach downwards to bundle near the symmetry axis. The stream ofbubbles is extended by some larger bubbles below the cone, but many smaller onesstop and merge near the cone tip. Bubble sizes (R0 of a few micrometer), velocities(about 1 m/s) and lifetimes (a few hundreds of driving periods or until collision withother bubbles) seem to correspond more or less to the data of stronger streamers.The sonotrode surface is partly covered by smoker and bubblyweb structures (seeSec. 3.9). For smaller sonotrodes with an artificially blocked liquid streaming at thesonotrode edges, conical bubble structures on a smaller scale have been reported[74].The acoustic field is mainly a propagating wave with an amplitude maximum closeto (but not exactly at) the sonotrode (see [78] for field calculations). The radiated

27

1 cm

Figure 17:Flare structure next to a submerged transducer of large area. The left part isattached to the vertical radiating surface, and the direction of bubble stream is to the right(scattered light, 25 kHz, exposure time 333µs).

intensities range from 1.5 to 8 W/cm2. Recordings with fixed phase relative to thedriving signal show mutual synchronous bubble oscillations, but drifting instants ofcollapse [78]. This is an indication for a distorted wave propagation, probably dueto nonlinear effects induced by high bubble density [79].The structure can be explained quite well by primary Bjerknes forces if the fullexpression (5) is taken into account and if a traveling wave of decreasing ampli-tude is assumed [80]. Following this explanation, the stagnation point at the conetip corresponds to a force equilibrium of the amplitude gradient part [first term inEq. (5)] and the phase gradient part (second term) of the primary Bjerknes force.The amplitude decline is caused by geometrical spreading ofthe wave front and bydissipation in the bubble field.Some phenomena that have been recorded during the switchingon of the transducerand for a pulsed excitation [78] remain to be explained, in particular the emergenceof larger bubble clusters prior to the full development of the conical structure.

3.8 Flare

This acoustic cavitation structure has been observed at large radiating surfaces likethe bottom of cleaning baths or submerged transducers. Froma limited part of thesurface (a few cm2), bubbles are ejected into the liquid, and they follow a broadpath which appears like aflare, see Fig. 17. The stream of bubbles wiggles anddevelops bumbs similar to a Kelvin-Helmholtz instability which is shown by freejet streams. Furthermore, the bubbly flare can detach from the surface and travela distance into the bulk liquid, whereupon it disappears. Itis composed of a fewlarge and many more small bubbles which form small dense filaments. The part

28

attached to the transducer appears reminiscent of the cone bubble structure, but theflare is extended far into the liquid beyond the cone tip or stagnation point, as canbe seen on the figure. On its way, the width of the bubbly zone grows, being fedfrom bubble sources at the side of its path. Bubble speeds of 1m/s are common,and the envelope of the whole structure is drifting with about 15 cm/s away fromthe transducer. Bubble sizes appear similar to streamer andjellyfish structures.The sound field in front of the large emitter has not been determined experimen-tally, but it probably possesses a complicated near field structure with shares ofboth traveling and standing waves. The amplitude distribution of the oscillation onthe surface itself is likely to be complex, because the back is equipped with severalindividual transducers. Nevertheless, the flare structureis universally found for dif-ferent large emitters, e.g. in various cleaning bath setups. The pressure amplitudesare supposed to be quite high, because at the same time jellyfish structures appearin other parts of the liquid container.The mechanism of the flare is unclear. The part near the emitter is possibly createdin a similar way as the CBS, but the jet of bubbles needs further explanation. Thesimilarity to a free jet is striking, in particular because of the shear-like instabilitiesat its boundaries. It remains to be explained at all why the boundaries of the bubblyphase are relatively well defined, in particular because theactual motion of thewater, induced by acoustic streaming, appears to be much smaller than the frontvelocity of the cavitating zone. Probably a significant interaction of propagatingsound wave and bubble density exists.

3.9 Smoker and web

This acoustic cavitation structure is bound to a surface in the liquid, and one exam-ple is seen in Fig. 18(a). It consists of an active spot on the object which is the originof a cloud or plume of very small bubbles. Because this plume is directed, and be-cause its constituents are hardly resolved, the optical appearance is reminiscent ofthe smoke emitted from a localized fire and blown away by the wind, watched fromthe top. Therefore the figure has been namedsmoker. It is always attached to asolid object, but the emission spot can be either fixed or drifting on its surface. Thebubble plume often runs close or parallel to the surface, andits direction can appearstationary or fluctuating. If more smokers are close to each other, their plumes canmerge. In extreme cases, merged plumes form awebor matrix of connected bubblyzones close to the object surface. Such a case is shown in Fig.18(b).Smokers appear on objects in strong acoustic fields. Preferred locations are trans-ducer surfaces, but the recording of Fig. 18(a) has been doneon a submerged alu-minium block in a standing wave bath. Multiple smokers can appear on passiveobjects, while really dense webs have been seen only directly on emitting surfaceslike in Fig. 18(b) which shows a large sonotrode surface generating a CBS (compare

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(b)(a)

Figure 18:Short exposure time recordings in reflected light of a smokerat 40 kHz (a) anda web at 20 kHz (b). Exposure times 2µs, picture heights about 1.5 cm in (a) and 3 cm in(b), material aluminium.

also [77]).On high-speed recordings, parts of the bubbles forming the smoker or web canvisually disappear during collapse. Furthermore, it has been observed that the plumecan oscillate in doubled or higher period [81], which means that the structure canemit subharmonics.The origin of the smoker is unclear. The smoker tips are strongly erosive points,and their location, if fixed, might be attached to local cracks on the surface. Smok-ers on materials other than aluminium, for example titanium, have been observed.Statistics about size and speed of the bubbles in the plume donot exist, but individ-ual bubbles have been measured to move with about 1 m/s away from the tip [81].Many bubbles are too small for optical resolution, and therefore they appear nebu-lous like a cloud. The boundary of such a cloud has been determined to move in therange of 10 m/s [81], but it is not sure if individual bubbles reach this speed, or if itis only a virtual speed of the front of bubbly liquid. The reason for the directivity ofthe bubble plume is also obscure. Liquid streaming is likelyto be too slow to haveany influence, but acoustic forces might be responsible. Themerging of smokerclouds to webs suggests that a certain attraction is acting between them.Once a transducer surface is covered by a dense bubble web, its emission proper-ties might change considerably and impedance mismatch, shielding and high localdissipation can appear. Nevertheless, the region close to the transducer can be veryactive and effective for certain processes [77].

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4 Summary

An introduction to the topic of acoustic cavitation and structure formation has beengiven, addressing aspects of sound fields and retroaction bycavitation, nucleationand bubble sources, cavitation thresholds, and Bjerknes forces.A variety of acoustic cavitation structures in the low frequency ultrasonic range hasbeen described, illustrated by images, and characterized up to available knowledge.Roughly they can be grouped into standing wave, traveling wave, and surface boundfigures, but this classifying scheme is not working perfectly. Structures connectedintimately with standing waves arestreamer filaments, bowsand rings, and thelayer structures (jellyfish andstarfish). On the other side,sonotrode, conical andflare bubble structures seem to rely strongly on traveling wave parts in the soundfield. Clustersof different sizes can appear in all kinds of acoustic fields,in thebulk liquid and on object surfaces.Smokersandwebsare definitely adherent tosolids (transducers, walls, or submerged objects). More types of bubble structuresare likely to be found and described in the future, but a reasonable classification intoa limited and small number of prototypes seems feasible.Distinctions of the proposed bubble structure candidates are obviously given in theiroptical shape and their acoustic environment. On the level of bubble dynamics,bubble sizes (equilibrium radii) of a few micrometer and below are quite commonand similar for all observed formations. As well, the measured translation speedsare more or less generally limited at about 1 m/s. Differences appear in the lifetimeof individual bubbles, the bubble number density, and the collapse strength. Theseparameters are probably the most significant ones to explaindifferent cavitationeffects under different conditions or setups.Lifetimes range from only one or few expansion and collapse cycles in clustersand below a sonotrode up to hundreds of cycles in streamers and filaments. Evenarbitrary many oscillations with strong collapse have beenrealized for levitated iso-lated bubbles [12]. Limiting factors of the lifetimes are high bubble density (orlow distance to a neighboring bubble, dominant for instancein clusters), collisionwith neighbors after spatial translation (dominant in streamer filaments), and sur-face instability after growth by rectified diffusion [38]. Probably the lifetime ofthe bubbles in different structures is explaining partly the variation of effects underdifferent acoustic conditions.The bubble density for many of the proposed situations varies strongly in space,which is the well known inhomogeneity of acoustic cavitation. Next to intenselycavitating regions, like in clusters, jellyfish layers, or flares, no bubbles can be no-ticed by optical means. Also the density within the populated regions of the struc-tures differs. Higher number densities can be found in layers, clusters, sonotrodestructures and flares, and in smoker plumes. Lower densitiesappear in filamentsof streamers and conical structures. The density should have an influence on the

31

sphericity of the bubbles, and therefore on both the bubble lifetime and collapsestrength. Also, the overall effect or yield of a cavitation application should scalewith the density.In the considered frequency range and for the applied sound field intensities, pre-sumably all active (i.e. strongly expanding and collapsing) bubbles are visible athigh optical resolution. Therefore, the observed bubbly regions should give an up-per estimate of the active sites. Of course, not all visible bubbles need to be stronglyimploding, and also the real violence of the collapse is difficult to assess. Bubblesdisappearing from the image during collapse likely emit shock waves, but even thenthe vehemence is unclear. It depends onR0 and on the local pressure amplitude,which might be quite different for different structures. Itcan be speculated that instanding waves the bubbles typically do not reach areas of pressure amplitude aboveabout 200 kPa when their structures are Bjerknes force controled [82]. In contrast,in traveling waves near sonotrodes the driving pressures are supposed to be signif-icantly higher. Then, however, the stronger collapse is accompanied by a shorterlifetime.

5 Outlook

This article has proposed classifications of bubble structures that represent differentmanifestations of acoustic cavitation. This has been done on the basis that, withrespect to this topic, not much information is available from prior publications, thatmany aspects are still unclear and that a general scheme of bubble figures has notbeen elaborated before. Being aware of the limitations of a review of this kind, itis intended that the information given will serve as a reference for forthcoming re-search. Future work needs to extend the results for a larger variety of acoustic andgeometric environments, as well as for higher and multiple excitation frequencies.Many aspects of the differences of bubble dynamics within the different structuresremain to be highlighted, as well as the consequences on distinct applications ofacoustic cavitation like cleaning or chemistry. An increasing amount of such ap-plications demands for a detailed knowledge of the link between acoustic field andcavitation effects, which is mediated by the distribution and properties of the cavi-tation bubbles.

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Acknowledgement

Many thanks go to all actual and former members of the cavitation group at the DPI inGottingen. Without their contributions, this article would not have been possible. I amindebted to Dagmar Krefting, Stefan Luther, Philipp Koch, Topi Tervo, Alexei Moussatovand Bertrand Dubus for making available pictures from shared or their own work. Specialthanks go to Claus-Dieter Ohl and Reinhard Geisler. I thank Werner Lauterborn for strongsupport, Juan Antonio Gallego-Juarez and Enrique Riera fortheir kind hospitality duringa stay at CSIC in Madrid, and Carsten Knieß from VKT Company for making availabletheir high-speed camera. Parts of this work have been supported by the German ScienceFoundation (DFG Graduiertenkolleg “Stromungsinstabilitaten und Turbulenz”), the Euro-pean Union (INCO Copernicus IC15-CT98-0141), and the German Ministry of Researchand Technology (BMBF project “Untersuchung von Kavitationsfeldern”).

References[1] D. H. Trevena,Cavitation and Tension in Liquids, Adam Hilger, Bristol, 1987.

[2] R. T. Knapp, J. W. Daily, & F. G. Hammitt,Cavitation, McGraw-Hill, New York, 1970.

[3] C. E. Brennen,Cavitation and Bubble Dynamics, Oxford University Press, New York, 1995.

[4] Y. Lecoffre,Cavitation Bubble Trackers, Balkema, Rotterdam, 1999.

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